Aircraft: MEDIUM TRANSPORT; LOW WING; 2 TURBOJET ENG
Phase of flight: Descent
Anomaly: ATC Issue All Types; Conflict Airborne Conflict; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy

I was the radar watch supervisor at the time of the incident; and was also working the flight data position.OJT was in progress at West Radar. Aircraft Y called Local Controller for taxi clearance; the pilot requested VFR departure to AVL at 4;500 feet; and Local Controller assigned 4;500 feet. Local Controller cleared the aircraft for takeoff. The pilot contacted West Radar; who advised the pilot to proceed on course; and stated 'Altitude your discretion.' The pilot then stated that he 'Would probably go up to about 10;000 feet.' This action put Aircraft Y opposite direction to and converging with Aircraft X; which was 28 miles away in AVL Approach airspace; landing GSP; and descending out of 14;000 feet for 7;000 feet. (West had already accepted a handoff from AVL on Aircraft X.)West called Aircraft X to Aircraft Y. (Aircraft Y was at 10;000 feet; and Aircraft X was descending through 11;000 feet; the aircraft were now 14 miles apart with a closure rate of 600 knots.) He then requested AVL to assign a left turn heading 110 to Aircraft X; AVL advised that he would once the aircraft descended into AVL airspace (upper limit 10;000 feet); and requested West to start Aircraft Y down. No descent was issued.Immediately after this I instructed West Radar to 'Do something to ensure they don't converge.' West then issued a left turn heading 290; and a subsequent turn to heading 270; but this was insufficient to prevent a TCAS RA by Aircraft X; who climbed from 10;100 feet to 10;600 feet before resuming descent. Before switching Aircraft X to West; AVL advised that the aircraft had responded to an RA. The closest lateral proximity was 2.55 miles.The following day I discussed the incident with the trainee; the OJTI; and the trainee's supervisor. Both the trainee and the OJTI believed that Aircraft X would descend below Aircraft Y without any problem. In addition; the OJTI thought Aircraft X was further east than it actually was. (Arrivals from that direction are usually on the UNMAN STAR; which was a few miles east of Aircraft X; it appeared that the aircraft was direct GSP.) The trainee had forgotten that Aircraft X needed to stop its descent at 10;000 feet in order to slow to 250 knots.1. A number of controllers at this facility; including the trainee in this incident; state 'Altitude your discretion' to VFR departures. This phraseology is not consistent with positive control. In this particular situation; the best course of action was to permit Aircraft Y to climb no higher than 6;000 feet; which would have eliminated any chance of a TCAS RA. Had the pilot wanted higher; and his initial request to the tower was 4;500 feet; he could have been given higher with a vector to ensure that Aircraft X was not a factor.2. It appears that many of our newer terminal radar controllers are not cognizant of closure rates; and simply do not realize just how fast two jets can converge with each other. This is an occasional problem on the southwest side of our airspace where we can have arrivals descending from 11;000 feet over ELW VOR with departures climbing to 10;000 feet towards the same NAVAID. During this incident the closure rate exceeded 600 knots; and when the trainee first called traffic; the aircraft were about 80 seconds away from a collision. Greater emphasis on closure rates during radar classrooms at the FAA Academy and at the facility would be beneficial. GSP TRACON Flight Line Manager (FLM) reports of a loss of separation by a training team on West radar. The controllers did not separate two aircraft one VFR climbing and one IFR descending.